Let $f : \mathbb Z\to \mathbb Z/x\mathbb Z \times \mathbb Z/y\mathbb Z$ be the homomorphism defined by $f (n) = (n + xZ, n + yZ)$...

Question

For $x,y \geq 2$, let $f : \mathbb Z\to \mathbb Z/x\mathbb Z \times \mathbb Z/y\mathbb Z$ be the ring homomorphism defined by $f (n) = (n + xZ, n + yZ)$.

(i) The kernel $K$ of $f$ is the ideal $s\mathbb Z$ for some number $s$ (depending, on $x$ and $y$). Describe $s$.

(ii) Show $(1+x\mathbb Z,0+y\mathbb Z)$ is in the image of $f$ iff $x$ and $y$ are coprime.

(iii) Decide which pairs $(a + x\mathbb Z, b + y\mathbb Z)$ are in the image of $f$. (Chinese Remainder Theorem is helpful for this)

Answer 1

Here's an answer to (i). To solve the other ones, try translating the jargon into a statement about integers and divisibility, or something that is more familiar or intuitive to you.

(i) You have $n \in Ker(f)$ if and only if $n + xZ = 0 + xZ$ and $n + yZ = 0 + yZ$, if and only if $x$ and $y$ both divide $n$. But $x$ and $y$ both divide $n$ if and only if $n$ is divisible by the least common multiple of $x$ and $y$, which we may call $m$.

Thus, $Ker(f) = m \mathbb{Z}$ (note that $m \mathbb{Z}$ is literally the set of integers which are divisible by $m$).

Answer 2

Let $\phi :\mathbb{Z}\rightarrow\mathbb{Z}_x\times\mathbb{Z}_y$. where is defined as $\phi(a)=(a+x\mathbb{Z}\cap, a+ y\mathbb{Z})$.

Then $ker(\phi)=x\mathbb{Z}\cap y\mathbb{Z}=l.c.m[x,y]\mathbb{Z}$.

So, if $(x,y)=1$, then $x\mathbb{Z}\cap y\mathbb{Z}=xy\mathbb{Z}$.

Then there exist $m,n\in \mathbb{Z}$ such that $mx+ny=1$.

Then $ny =1- mx$ Then $\phi: ny \to (0+y\mathbb{Z})$, and $(1-mx)\to ((1+x\mathbb{Z})$. So then $(1+x\mathbb{Z},0+y\mathbb{Z})\in im(\phi)$.

Then $(a+x\mathbb{Z},b+y\mathbb{Z})\in im(\phi)$ when $\exists d \in \mathbb{Z}$ such that $d\equiv a (mod\ x)$ and $d\equiv b (mod\ y)$. The $\phi(d)=(a+x\mathbb{Z},b+y\mathbb{Z})$.